Is there a way to put a copy rotation constraint with an influence higher than 1?
1.0 is the maximum influence for any given constraint, so setting it beyond 1.0 is impossible.

Using the Transformation constraint, you could easily map any kind of simple transformation from one object to the other. In the following example, the object being constrained will rotate twice the amount of Cube_target's rotation in the Y axis.



Checking Extrapolation will extend the transformation beyond stated range of values.